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Volume 39, Issue 4
Error Estimates for Two-Scale Composite Finite Element Approximations of Nonlinear Parabolic Equations

Tamal Pramanick

J. Comp. Math., 39 (2021), pp. 493-517.

Published online: 2021-05

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  • Abstract

We study spatially semidiscrete and fully discrete two-scale composite finite element method for approximations of the nonlinear parabolic equations with homogeneous Dirichlet boundary conditions in a convex polygonal domain in the plane. This new class of finite elements, which is called composite finite elements, was first introduced by Hackbusch and Sauter [Numer. Math., 75 (1997), pp. 447-472] for the approximation of partial differential equations on domains with complicated geometry. The aim of this paper is to introduce an efficient numerical method which gives a lower dimensional approach for solving partial differential equations by domain discretization method. The composite finite element method introduces two-scale grid for discretization of the domain, the coarse-scale and the fine-scale grid with the degrees of freedom lies on the coarse-scale grid only. While the fine-scale grid is used to resolve the Dirichlet boundary condition, the dimension of the finite element space depends only on the coarse-scale grid. As a consequence, the resulting linear system will have a fewer number of unknowns. A continuous, piecewise linear composite finite element space is employed for the space discretization whereas the time discretization is based on both the backward Euler and the Crank-Nicolson methods. We have derived the error estimates in the $L^\infty(L^2)$-norm for both semidiscrete and fully discrete schemes. Moreover, numerical simulations show that the proposed method is an efficient method to provide a good approximate solution.

  • AMS Subject Headings

35J20, 65N15, 65N30

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

t.pramanick002@gmail.com (Tamal Pramanick)

  • BibTex
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@Article{JCM-39-493, author = {Pramanick , Tamal}, title = {Error Estimates for Two-Scale Composite Finite Element Approximations of Nonlinear Parabolic Equations}, journal = {Journal of Computational Mathematics}, year = {2021}, volume = {39}, number = {4}, pages = {493--517}, abstract = {

We study spatially semidiscrete and fully discrete two-scale composite finite element method for approximations of the nonlinear parabolic equations with homogeneous Dirichlet boundary conditions in a convex polygonal domain in the plane. This new class of finite elements, which is called composite finite elements, was first introduced by Hackbusch and Sauter [Numer. Math., 75 (1997), pp. 447-472] for the approximation of partial differential equations on domains with complicated geometry. The aim of this paper is to introduce an efficient numerical method which gives a lower dimensional approach for solving partial differential equations by domain discretization method. The composite finite element method introduces two-scale grid for discretization of the domain, the coarse-scale and the fine-scale grid with the degrees of freedom lies on the coarse-scale grid only. While the fine-scale grid is used to resolve the Dirichlet boundary condition, the dimension of the finite element space depends only on the coarse-scale grid. As a consequence, the resulting linear system will have a fewer number of unknowns. A continuous, piecewise linear composite finite element space is employed for the space discretization whereas the time discretization is based on both the backward Euler and the Crank-Nicolson methods. We have derived the error estimates in the $L^\infty(L^2)$-norm for both semidiscrete and fully discrete schemes. Moreover, numerical simulations show that the proposed method is an efficient method to provide a good approximate solution.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2001-m2019-0117}, url = {http://global-sci.org/intro/article_detail/jcm/19148.html} }
TY - JOUR T1 - Error Estimates for Two-Scale Composite Finite Element Approximations of Nonlinear Parabolic Equations AU - Pramanick , Tamal JO - Journal of Computational Mathematics VL - 4 SP - 493 EP - 517 PY - 2021 DA - 2021/05 SN - 39 DO - http://doi.org/10.4208/jcm.2001-m2019-0117 UR - https://global-sci.org/intro/article_detail/jcm/19148.html KW - Composite finite elements, Nonlinear parabolic problems, Coarse-scale, Fine-scale, Semidiscrete, Fully discrete, Error estimate. AB -

We study spatially semidiscrete and fully discrete two-scale composite finite element method for approximations of the nonlinear parabolic equations with homogeneous Dirichlet boundary conditions in a convex polygonal domain in the plane. This new class of finite elements, which is called composite finite elements, was first introduced by Hackbusch and Sauter [Numer. Math., 75 (1997), pp. 447-472] for the approximation of partial differential equations on domains with complicated geometry. The aim of this paper is to introduce an efficient numerical method which gives a lower dimensional approach for solving partial differential equations by domain discretization method. The composite finite element method introduces two-scale grid for discretization of the domain, the coarse-scale and the fine-scale grid with the degrees of freedom lies on the coarse-scale grid only. While the fine-scale grid is used to resolve the Dirichlet boundary condition, the dimension of the finite element space depends only on the coarse-scale grid. As a consequence, the resulting linear system will have a fewer number of unknowns. A continuous, piecewise linear composite finite element space is employed for the space discretization whereas the time discretization is based on both the backward Euler and the Crank-Nicolson methods. We have derived the error estimates in the $L^\infty(L^2)$-norm for both semidiscrete and fully discrete schemes. Moreover, numerical simulations show that the proposed method is an efficient method to provide a good approximate solution.

Tamal Pramanick. (2021). Error Estimates for Two-Scale Composite Finite Element Approximations of Nonlinear Parabolic Equations. Journal of Computational Mathematics. 39 (4). 493-517. doi:10.4208/jcm.2001-m2019-0117
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